 Darcy velocity

Darcy velocity, also called Darcy flux or specific discharge, is fluid volume flux relative to the solid framework per unit total cross-sectional area. For a uniform axial discharge $Q$ through bulk area $A$,The area includes both solid and voids. Hence Darcy velocity is a superficial flux with units $\mathrm{m}{\mathrm{s}}^{-1}$, even though no fluid travels through the solid part. For a nonuniform discharge, $Q={\int }_A\mathbf{q}\cdot \mathbf{n}dA$, with $\mathbf{n}$ the unit normal; $Q/A$ is the area average of the normal Darcy flux.

If porosity is $ϵ$, the pore-volume-averaged liquid velocity is ${\mathbf{v}}_l$, and the local solid framework velocity is ${\mathbf{v}}_s$, the convention used here is $\mathbf{u}=ϵ({\mathbf{v}}_l-{\mathbf{v}}_s)$. In a stationary, fully saturated framework with all pore volume participating, ${\mathbf{v}}_l=\mathbf{q}/ϵ$. The notation $\varphi =ϵ$ for porosity is equally common. Darcy flow through a homogeneous slab gives $q=2.0\times 10^{-5}\mathrm{m}{\mathrm{s}}^{-1}$ but pore velocity $v=8.0\times 10^{-5}\mathrm{m}{\mathrm{s}}^{-1}$ at $\varphi =0.25$. Keeping these quantities separate lets Darcy law predict filter discharge while pore velocity predicts ideal advective travel time.^[ref]